A toothed wheel of diameter $50 cm$ is attached to a smaller wheel of diameter $30\text{ cm}$.Then the number of revolutions the smaller wheel will make when the larger one makes $30$ revolution is

When two toothed wheels are joined together, the distance travelled by one wheel after n revolutions is equal to the distance travelled after m revolutions by the second wheel. The diameter of the circular wheel determines how far it travels in one rotation. Since πd is the diameter of the circular wheel, we know that the circumference of a circle equals d.
In light of this, a wheel with 30 cm of teeth and a 50 cm diameter toothed wheel are mounted.
The circumference of a circular wheel with a 50cm diameter indicates the amount of distance it travels. Consequently, the circle's circumference is 50 cm.

$D_1=\pi d$ $=50\pi$
The circumference of a circular wheel with a diameter of 30 cm indicates the distance it travels. As a result, the circle's 30 cm circumference is
$D_2=\pi d$
$=30\pi$
Given that the huge circle, with a diameter of 50 cm, will complete 30 revolutions, the distance it will go is
$D_n=30\left(D_1\right)$
$=30\left(50\pi \right)$
Let's now assume that the 30 cm tiny tooth wheel travels the distance $D_n$ This, in k rotations, is produced by a bigger wheel with a diameter of 50 cm. The distance traveled for k rotations by the tiny wheel with a 30 cm diameter is thus
$D_k=k\left(D_2\right)$
$=k\left(30\pi \right)$

Now we have $D_n=D_k$, substituting both the values in the above equation then
$30\left(50\pi \right)=k\left(30\pi \right)$
$k=\frac{30\left(50\pi \right)}{30\pi }$
$k=50$

Thus, a wheel with a diameter of 30 cm will turn 50 times, but a wheel with a diameter of 50 cm will turn 100 times.

Hence, the correct answer is option B.